The present invention relates to the art of fitting kinetic models. In particular, it relates to fitting mathematical models that describe how the behavior of a drug or substance (herein referred to as a “tracer”) changes as a function of time in a physiologic system to measurements of such dynamic processes. The task of fitting kinetic models arises in a wide variety of fields, including but not limited to drug development, preclinical and diagnostic imaging, and radiation dosimetry.
The prototypical kinetic model is a compartment model, which describes how the tracer (drug or substance) exchanges between the input source (typically arterial blood) and different tissue locations and/or different chemical forms occupying the same physical space (“compartments”). However, a number of other types of kinetic models exist, such as those that model the system as a sum of exponentials or other functions, often convolved with the input source function. In a typical application, a series of measurement of the tracer concentration are made sequentially over period of minutes, hours, or days, as the tracer distributes physiologically. Measurements of the tracer concentration in the source input (e.g. arterial blood) may also be made in some cases. These data in aggregate measure how the tracer concentration changes as a function of time—i.e., a dynamic curve. A mathematical model that predicts the dynamic curve in terms of meaningful and quantifiable parameters is then fit to the measured data, providing estimates of the model parameters that relate to the underlying physiologic processes controlling the tracer distribution. These model parameters are the desired endpoints of the kinetic model fitting process.
The nature of how tracers distribute in physiologic systems gives rise to complex modeling equations that include “nonlinear” terms. Such nonlinear equations are much more difficult to solve than linear equations, which can often be solved analytically in a single step. Mathematical fits to nonlinear equations cannot be solved in a single step, but instead require that a number of possible solutions be tested in order to find the solution that most-closely matches the measured data.
The current state of the art in kinetic modeling is to use iterative nonlinear fitting algorithms to minimize a least squares-type objective function, providing an estimate of the least-squares (or weighted least-squares) solution. These nonlinear least-squares (NLLS) algorithms are often considered to provide the most robust kinetic model fits, but have a number of shortcomings. The multi-dimensional nonlinear fitting space is complex and variable, including local minima, ridges, and valleys that can confound the iterative fitting algorithm. As a result, the NLLS solution is dependent on the initial conditions provided by the user, requiring in many cases that the user manually provide a set of initial conditions that is already an approximate match for the measured dynamic curve being fit. The number of iterations and iterative stopping criterion need to be carefully managed. Once the fit is obtained, careful attention to quality assurance must be performed in order to ensure that the resultant fit closely-matches the measured data; even then, however, one can never be certain that the true least-squares best fit has been obtained.
More robust versions of NLLS fitting algorithms have been tested, which resort to restarting the fits over and over again with changing conditions in order to escape local minima and reduce the dependency on initial conditions. Such approaches, however, tend to be very computationally demanding. Alternative algorithms, such as simulated annealing, use related schemes to increase the chances of obtaining the true global minimum fit. However, these are also very computational demanding, requiring extensive computer power and time for each fit. A variety of other approaches have been and are under development, including a variety of linearized fitting approaches, graphical analysis methods, and basis function approaches. While these methods show promise for some applications, each makes certain approximations or has other complicating factors that fall short of the ideal fitting solution.
These earlier works have not resulted in a kinetic model fitting approach that is simple, fast, effective, and robustly finds the true best-fit solution in all cases in the presence of real-world complicating factors such as high levels of noise in the measurements.